Use the Pythagorean Theorem: \begin{align*}a^2+b^2=c^2\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are the lengths of the legs and \begin{align*}c\end{align*} is the length of the hypotenuse. When we substitute the values from the diagram, we get \begin{align*}x^2+(x+3)^2=15^2\end{align*}.

In order to solve this equation, we need to get the polynomial in standard form. We must first distribute, collect like terms and rewrite in the form “polynomial = 0.”

Factor out the common monomial: \begin{align*}2(x^2+3x-108)=0\end{align*}

To factor the trinomial inside the parentheses, we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be \begin{align*}x = 9\end{align*}. That means the short leg is 9 feet and the long leg is 12 feet.

Check:\begin{align*}9^2+12^2=81+144=225=15^2\end{align*}, so the answer checks.

Real-World Application: Finding Unknown Numbers

The product of two positive numbers is 60. Find the two numbers if one number is 4 more than the other.

Let \begin{align*}x =\end{align*} one of the numbers; then \begin{align*}x + 4\end{align*} is the other number.

The product of these two numbers is 60, so we can write the equation \begin{align*}x(x+4)=60\end{align*}.

In order to solve we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

\begin{align*}x^2+4x &= 60\\
x^2+4x-60 &= 0\end{align*}

Factor by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:

Example

Example 1

Consider the rectangle in Example C with sides of length \begin{align*}x + 5\end{align*} and \begin{align*}x - 3\end{align*}. What is \begin{align*}x\end{align*} if the area of the rectangle is now 20?

Make a sketch of this situation:

Using the formula Area = length \begin{align*}\times\end{align*} width, we have \begin{align*}(x+5)(x-3)=20\end{align*}.

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

\begin{align*}x^2+2x-15& =20\\
x^2+2x-35& =0\end{align*}

Factor by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:

Review

One leg of a right triangle is 1 feet longer than the other leg. The hypotenuse is 5. Find the dimensions of the right triangle.

One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.

A rectangle has sides of \begin{align*}x + 2\end{align*} and \begin{align*}x - 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 108?

A rectangle has sides of \begin{align*}x - 1\end{align*} and \begin{align*}x + 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 120?

The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.

A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?

Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?

Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?

Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs $2 per foot. If the frame has to be a square, what size picture can you get framed for $20?

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.14.